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Full text of "Treatise On Analysis Vol-Ii"

294       XIV    INTEGRATION IN LOCALLY COMPACT GROUPS

(ii) Ifp = 1 or 2 and ge JS?P(G), the sequence of norms Np((/n * g) - g)
tends to 0 as n tends to + 00. For each function g e Lg3(G), the sequence
((fn *#)~) converges weakly to g in L(G), considered as the dual of L(G)
(12.15 and 13.17).

(iii) Suppose in addition that the supports of the fn are contained in a
fixed compact subset of G, Then for each measure \JL on G, the sequence of
measures

converges vaguely to /* (13.4).

(i)   For each x e G and each compact neighborhood V of e we have
from the definitions

- f /.(

f/

-i

Cv

Let V0 be a compact neighborhood of e and let L be a compact subset of
G. Since V^"1L is compact (12.10.5), the restriction of g to VQ JL is uniformly
continuous with respect to a right-invariant distance on G (3.16.5). Hence,
for each e > 0, there exists a compact neighborhood V c V0 of e such that
\ff(x)-~9($~ix)\^ for all xeL and all seV. Now choose n0 so that
JCy \ftt(s)\ dp(s) ^ e and |1 - jfn(s) dp(s)\ g e for all n ^ n0. We have then

1-

and therefore, for all x e L,

e,

Cv

where ^ = sup ) \ftt($)\ dft(s). Hence

' with respect